Express $z_1=5\left[\cos\left(\dfrac{7\pi}{4}\right)+i\sin\left(\dfrac{7\pi}{4}\right)\right]$ in rectangular form. Express your answer in exact terms. $z_1=$
Explanation: The Strategy A complex number in rectangular form, $z={a}+{b}i$, can be written in polar form as $z={r}[\cos{\theta}+i\sin{\theta}]$, where ${r}$ is the absolute value, or modulus, and ${\theta}$ is the angle, or argument. Therefore, ${r}$ and ${\theta}$ can be found using the following formulas: ${r}=\sqrt{{a}^2+{b}^2}$ $\tan{\theta}=\dfrac{{b}}{{a}}$ [How did we get these equations?] Similarly, a complex number in polar form, $z={r}[\cos{\theta}+i\sin{\theta}]$, can be written in rectangular form as $z={a}+{b}i$, using the following formulas: ${a}={r}\cos{\theta}$ ${b}={r}\sin{\theta}$ [How did we get these equations?] Finding $a$ For $z_1={5}[\cos{\dfrac{7\pi}{4}}+i\sin{\dfrac{7\pi}{4}}]$ : ${r}={5}$ ${\theta}={\dfrac{7\pi}{4}}$ Therefore, we can find ${a}$ as follows. $\begin{aligned}{a}&={r}\cos{\theta} \\\\&={5}\cos{\dfrac{7\pi}{4}} \\\\&={\dfrac{5\sqrt{2}}{2}}\end{aligned}$ Finding $b$ $\begin{aligned}{b}&={r}\sin{\theta} \\\\&={5}\sin{\dfrac{7\pi}{4}} \\\\&={-\dfrac{5\sqrt{2}}{2}}\end{aligned}$ Summary $z_1={\dfrac{5\sqrt{2}}{2}}{-\dfrac{5\sqrt{2}}{2}}i$